Noise Traders, Exchange Rate Disconnect Puzzle, and the Tobin Tax

Transcription

1 Noise Traders, Exchange Rate Disconnect Puzzle, and the Tobin Tax September 2008 Abstract This paper proposes a framework to explain why the nominal and real exchange rates are highly volatile and seem to be disconnected from macroeconomic fundamentals. Two types of foreign exchange traders, rational traders and noise traders with erroneous stochastic beliefs, are introduced into a sticky-price dynamic general-equilibrium framework. The presence of noise traders creates deviations from the uncovered interest parity. As a result, exchange rates can diverge significantly from the fundamental values. Combined with local currency pricing and consumption smoothing behavior in an infinite horizon model, the presence of noise traders can help to explain the exchange rate disconnect puzzle. Then we show that the excess exchange rate volatility caused by the presence of noise traders can be reduced by the Tobin tax type of policies. However, the effect of the Tobin tax on the exchange rate volatility depends on the market structure of foreign exchange market and the interaction of the Tobin tax with other trading costs. JEL classification numbers: F41, F31, G15 Keywords: Noise traders; microstructure; exchange rate disconnect; macroeconomic fundamentals; volatility.

2 1 Introduction A central puzzle in international macroeconomics over the last 20 years is that real exchange rates are volatile and persistent. Furthermore, as Flood and Rose (1995) have elegantly documented, the exchange rate seems to have a life of its own, being disconnected from other macroeconomic variables. For example, Mussa (1986), Baxter and Stockman (1989) and Flood and Rose (1995) all find that both nominal and real exchange rates are highly volatile, especially when compared to the macroeconomic fundamentals, such as relative price level, consumption, and output. Exchange rate volatility also varies substantially over time. Obstfeld and Rogoff (2000) state this kind of exceedingly weak relationship between the exchange rate and virtually any macroeconomic aggregates as the exchange rate disconnect puzzle. This irregularity casts some doubts on the traditional monetary macroeconomic model of exchange rates, which assumes that purchasing power parity (PPP) holds. With PPP, the expenditure-switching effect of exchange rate changes will lead to substitution between domestically-produced goods and internationally-produced goods. It implies that the exchange rate volatility will be transferred to macroeconomic fundamentals. Nevertheless, empirical evidence 1 indicates that nominal exchange rate changes are not fully passed through to goods prices. Motivated by this evidence, Betts and Devereux (1996, 2000) introduce local currency pricing into the baseline Redux model developed by Obstfeld and Rogoff (1995). They assume that firms can charge different prices for the same goods in home and foreign markets and that the prices are sticky in each country in terms of the local currency. This allows the real exchange rate to fluctuate, and delinks the home and foreign price levels. Although the new open economy macroeconomic models with sticky prices, imperfect competition and local currency pricing can generate volatile exchange rate movements, 2 they typically predict a strong counterfactual relationship between the real exchange rate and relative consumption. A monetary shock simultaneously raises domestic consumption (by more than it raises foreign consumption) and creates a (temporary) depreciation of home currency. Consequently, these models almost generically predict a strong positive correlation between depreciation and relative consumption, which is not observed empirically. 3 For example, Chari, Kehoe and McGrattan (2002) refer to this puzzle as the consumption-real exchange rate anomaly and 1 See Engel (1993, 1999) and Parsley and Wei (2001) for details. 2 A high risk aversion coefficient of household (about 5) is usually required in these models to reproduce the data s volatility of real exchange rate relative to output. See Chari, Kehoe and McGrattan (2002). 3 Benigno and Thoenissen (2003) report the correlation between bilateral exchange rate and bilateral relative consumption for seven countries (Canada, France, West Germany, Italy, Japan, U.K. and U.S.) for the periods starting from 1970 until The cross-correlation varies between 0.45 and

3 they show that neither incomplete financial market nor habit persistence can eliminate it. One explanation for this discrepancy might lie in the fact that the nominal exchange rate is also an asset price, and therefore will be inevitably affected by imperfections in the financial markets. These imperfections may include herd behavior, momentum investing and noise trading. Working together with sticky prices, these are all important reasons to explain why the real exchange rate persistently deviates from the level predicted by the fundamentals-based models. Since the early nineties, many economists studied the behavior of markets participants in foreign exchange markets through survey conducted in dealers. 4 The evidence collected in these surveys showed that a majority of traders does not rely on information about fundamentals factors, but use various sources of technical information such as trend curves. Also, most traders believe that at least over the short and medium run, exchange rates are governed more by speculative behavior or technical trading rather than macroeconomic fundamental and pertinent news. Evans and Lyons (2002) show that most of the short-run exchange rate volatility is related to order flow, which also reflects the heterogeneity in investors expectations. Cai, Cheung, Lee and Melvin(2001) also provide evidence in support of the independent role of order flow and its associated information as a determinant of exchange rate dynamics. These pieces of evidence all suggest deviation from rational expectation and the extensive use of non-fundamental trading strategy in foreign exchange markets. Therefore, our paper intends to propose a new approach to study exchange rates, that combines the macroeconomic model of exchange rates and the microstructure approach of foreign exchange markets. This approach is implemented within a specific model, where noise traders are introduced into the new open economy macroeconomic framework. The combination is helpful for understanding the behavior of exchange rates and their relationship with macroeconomic fundamentals. It also gives more rigorous microeconomic foundations to the noise trader approach and enriches the new open economy macroeconomic framework with a more realistic setting of the microstructure of foreign exchange market. In addition, it provides a well-defined framework for policy evaluations, especially for policies designed to control non-fundamental volatilities, such as the Tobin tax. We adopt the overlapping-generation noise trader model of De Long et al. (1990). Two types of foreign exchange traders are introduced into the general equilibrium framework. One type is the rational/informed trader, which has rational expectations about future investment returns, while the other type cannot forecast the future returns correctly and is called the noise trader. 4 See Frankel and Froot (1987,1990), Allen and Taylor (1990), Taylor and Allen (1992), Cheung and Chinn (2001), Chinn and Frankel (1994), and Cheung and Wong (2000) for details. 2

4 The results from the model show that when the number of noise traders increases, so does the exchange rate volatility. Nevertheless, the volatilities of macroeconomic fundamentals (except for the net foreign assets) are completely independent of the noise component on the foreign exchange market. Therefore, our model can generate a relative volatility of real exchange rate to output close to the data, even for a low risk aversion coefficient. Moreover, since in this model nominal and real exchange rate fluctuations can be generated by erroneous belief of noise traders, our model does not predict a strong comovement of exchange rates and fundamentals. Therefore, it is possible to explain the exchange rate disconnect puzzle by the approach suggested in this paper. The basic intuition behind our results is as follows. The heterogeneity in beliefs among foreign exchange traders creates the basis for trading volume and deviations from the uncovered interest parity. This breaks the link between an exchange rate shock and the home relative real interest rates (home relative expected future consumption) implied by the interest rate parity condition. Therefore, exchange rates can diverge significantly from the fundamental values. The greater the number of noise traders, the more volatile will be the exchange rates. However, why is the exchange rate volatility not transferred to macroeconomic fundamentals? Normally, there are two channels through which the exchange rate affects the macroeconomic variables: the expenditure-switching effect and the wealth effect. Under the assumption of local currency pricing, the expenditure-switching effect is eliminated as the relative price of home and foreign goods does not change. Although the wealth effect still exists, it turns out to be quite small quantitatively. This is because the wealth effect of exchange rate change is spread out over current and future periods through intertemporal consumption smoothing, and so tends to be very small. Another important implication of our model is related to the consumption-real exchange rate anomaly. We show that we can get a much smaller cross-correlation between exchange rates and relative consumption in our model. Moreover, this cross-correlation decreases when more noise traders are present on the foreign exchange market. Intuitively, this is because our model setting separates the foreign exchange market from the rest of the model, insuring that the expectation error creates a source of fluctuation which only affects exchange rats, but not macroeconomic fundamentals. In other words, both expectation error shocks and monetary shocks cause exchange rate fluctuations, but macroeconomic fundamentals can only be affected by monetary shocks. So our model will not predict a strong comovement of the exchange rate and relative consumption. Many economists have suggested that increasing the trading cost on the foreign exchange market might reduce the exchange rate volatility. To understand the effect of this kind of 3

5 policies, the size of the noise component is endogenized by introducing a heterogenous entry cost for noise traders. Only noise traders having entry costs that are sufficiently low will choose to enter the foreign exchange market. We find that increasing the entry cost will reduce the exchange rate volatility. We also analyze the Tobin tax type of policy suggested by Tobin (1978) and Eichengreen, Tobin and Wyplosz (1995). We find that a Tobin tax will decrease the exchange rate volatility, however, the impact of a Tobin tax on exchange rate volatility depends crucially on the structure of the foreign exchange market and the interaction of the Tobin tax with other trading costs. One important policy implication given the model structure is that, Tobin tax and other kinds of entry cost can reduce the excess volatility without affecting the volatilities of key macroeconomics variables, such as consumption and output. This paper belongs to the new open economy macroeconomics literature. Wang (2008) demonstrates in a two-country DSGE model, that the home bias in consumption is important to duplicate the exchange rate volatility and exchange rate disconnect documented in the data. When home bias is high, the shock to uncovered interest rate parity can substantially drive up exchange rate volatility while leaving the volatility of real macroeconomic variables almost untouched. The paper that is closest, in spirit, to our analysis of exchange rate disconnect puzzle is Devereux and Engel (2002). They stated that the key ingredients for a general equilibrium model to explain the exchange rate disconnect puzzle include: local currency pricing to eliminate the expenditure-switching effect, a special structure of international pricing and product distribution to minimize the wealth effect, incomplete international financial markets, and stochastic deviations from the uncovered interest parity. The major difference between our paper and theirs is that we introduce more microeconomic foundations of noise traders. In our model, both noise traders and rational traders are risk averse and utility maximizing agents. Therefore, our paper provides a framework to analyze the implication of policies like Tobin tax. Also, instead of assuming a specific production and distribution structure to remove wealth effects completely, we show that the wealth effect is quite small quantitatively in a standard infinite horizon model. Finally, we point out the implication of expectation errors in explaining the consumption-real exchange rate anomaly. The microstructure of the exchange rate market in this paper follows the noise trader literature initiated by De Long et al. (1990), especially the work of Jeanne and Rose (2002). They also use noise traders to generate non-fundamental based exchange rate volatility. Their paper shows that for the same level of fundamental macroeconomic volatility, there exist multiple equilibria under floating regime. When noise traders are present, exchange rate volatility will be high; while when they are absent from the markets, exchange rate volatility is low. Although they focus on the the role of fixed exchange rate regime in reducing non-fundamental exchange 4

6 rate volatility, their results can also be used to explain the exchange rate disconnect puzzle. The macroeconomic part of their model, however, is a simple monetary model of exchange rates with PPP. Neither nominal rigidities nor pricing to market is considered. Moreover, intertemporal optimizing agents and profit maximizing firms are not modeled. So most channels through which the exchange rate affects macroeconomic fundamentals, such as the expenditure-switching effects and wealth effects of exchange rate changes, cannot be analyzed. Therefore, to study the exchange rate disconnect puzzle more rigorously, this model generalized the macroeconomic part of their model along the lines of the New Open Macroeconomics. Another feature of their model is that it is a partial equilibrium model without explicit welfare specifications for households, so rigorous policy analysis is impossible. 5 As to the discussion of the Tobin tax, our study contributes to the literature on effects of the Tobin tax on foreign exchange rate volatility. Since James Tobin propose the Tobin tax in 1974, the debate about the Tobin tax concentrates on it feasibility and the distorting effects it might have as a tax. Recently, some scholars question the conventional wisdom that an increase in the Tobin tax reduces market volatility. For example, see Davidson (1997, 1998) and Song and Zhang (2005). The major difference of our paper is that we study the effect of the Tobin tax on the volatility of exchange rates and macroeconomic fundamentals in a DSGE model. Also, we show that although a Tobin tax can reduce exchange rate volatility, its effect depends on the market microstructure and the interaction of the Tobin tax with other trading costs. This paper is organized as follows. In Section 2, we construct a model that embeds noise traders and the Tobin tax into a new open economy macroeconomic framework. Both the exogenous entry and endogenous entry specifications are explored. In Section 3 features of the solution to the model are discussed. Section 4 gives the results of the model. The paper concludes with a brief summary and suggestions for subsequent research. 5 Becchetta and van Wincoop (2004) and Evans and Lyons (2005) introduce order flow and information dispersion about future fundamentals into dynamic macroeconomic model to explain the exchange rate disconnect puzzle. There are several important differences in comparison to our approach. First, our paper follows the NOEM framework, while the macroeconomic framework in their papers is either a partial-equilibrium monetary model (Becchetta and van Wincoop, 2004) or a DGE model without nominal rigidities (Evans and Lyons, 2005), so the monetary policy analysis is not possible. Also, a lot of channels through which exchange rates affect macroeconomic fundamentals are not considered. Second, they focus on information dispersion, and our paper focuses on noise traders. Third, both papers emphasize the role of order flow in the information dispersion. Therefore, their settings of foreign exchange market are different from ours. 5

7 2 The model The world economy consists of two countries, denominated by home and foreign. Each country specializes in the production of a composite traded good. Variables in the foreign country are denoted by an asterisk. In addition, a subscript h denotes a variable originating from the home country; a subscript f denotes a variable produced in the foreign country. This model is analogous to most new open economy macroeconomic models except for the foreign exchange market. Each country is populated by a large number of atomistic households, a continuum of firms that set prices in advance, and a government (a combined fiscal and monetary authority). However, we assume that home and foreign households can only trade nominal bonds denominated in their domestic currency. Although home households cannot access the international bond market, the foreign exchange traders can carry out the international bond trading to maximize their utility. Thus, besides the infinitely lived household, a second type of representative agent is introduced into the model, namely, the foreign exchange trader, who lives in an overlapping-generation demographic structure. Hereafter, a superscript H denotes households and a superscript T stands for traders. In the foreign country, only one type of representative agent is present; the foreign household Households, Firms and Government The lifetime expected utility of the home representative household is: max E 0 { t=0 β t [ (C H t ) 1 ρ 1 ρ ɛ ( Mt P t ) 1 ɛ η ]} 1 + ψ L1+ψ t (2.1) Subject to P t C H t + B t+1 + M t = W t L t + Π t + M t 1 + T t + B t (1 + r t ) (2.2) where Ct H is the time t composite consumption of home households, composed by a continuum of home goods and foreign goods; both are of measure 1. Let Ct T denote the composite consumption of traders, then Ct T + Ct H = C t, where C t is the composite consumption of the home country and is given by: [ C t = ω 1 γ C γ 1 γ h,t + (1 ω) 1 γ C γ 1 ] γ γ 1 γ f,t (2.3) 6 We can make the model more symmetric by assuming half of the traders lives in the home country and half of them lives in the foreign country. But this will only complicate the model without significantly changing the results. 6

8 where C h,t = ( ) 1 0 C h,t(i) θ 1 θ θ di θ 1, C f,t = ( ) 1 0 C f,t(j) θ 1 θ θ dj θ 1, and the weight ω (0, 1) determines the home representative agent s bias for the domestic composite good. Note that θ is the elasticity of substitution between individual home(or foreign) goods and γ is the elasticity of substitution across home and foreign composite goods. P t is a consumption based price index for period t, which is defined by: [ P t = ωp 1 γ h,t + (1 ω)p 1 γ ] 1 1 γ f,t where P h,t = ( 1 0 P h,t(i) di) 1 θ 1 ( 1 θ and P f,t = 1 0 P f,t(j) dj) 1 θ 1 1 θ. (2.4) In each period every household is endowed with one unit of time, which is divided between leisure and work. His income is derived from the labor income W t L t, profits from domestic goods producers (which is assumed to be owned by domestic households) Π t, interest received on domestic bonds B t (1 + r t ) and lump-sum government transfer T t. Solving the household s problem, the optimality conditions can be written as: ) ɛ = (CH t ) ρ ( Mt P t ηl ψ t = βe t (C H t ) ρ (C H t+1 )ρ P t = r t+1 (2.5) W t P t (C H t )ρ (2.6) r t+1 (2.7) The first order conditions of the foreign households are entirely analogous, except that foreign household s consumption is denoted by C t, as there is only one type of representative agent in the foreign country. We assume firms have linear technologies, for each home good i, y t (i) = L t (i). It is also assumed that, due to high costs of arbitrage for consumers, each individual monopolist can price discriminate across countries. Furthermore, as in Betts and Devereux (1996) and Chari, Kehoe and McGrattan (2002), we assume local currency pricing: firms set prices (separately) in the currencies of buyers. Finally, prices are assumed to be set one period in advance and cannot be revised until the following period. That is, the home monopolist sets P h,t (i) and P h,t (i) optimally at the end of period t 1, and these prices cannot be changed during time t. The technical appendix gives the derivation of the optimal pricing schedule of firms. The firms will just set the price so that it equals to a mark-up over the expected marginal cost and a risk premium term arising from the covariance of the firm s profits with the marginal utility of consumption: P h,t = θ E t 1 [D t W t C t ] θ 1 E t 1 [D t C t ] 7 P h,t = θ E t 1 [D t W t Ct ] θ 1 E t 1 [D t S t Ct ] (2.8)

9 P f,t = θ E t 1 [Dt Wt C t ] [ θ 1 D ] Pf,t = θ E t 1 [Dt Wt Ct ] E t t 1 S t C t θ 1 E t 1 [Dt C t ] (2.9) where D t and D t denote the pricing kernels households used to value date t profits. Because all home firms are assumed to be owned by the domestic households, it follows that in equilibrium D t is the intertemporal marginal rate of substitution in consumption between time t 1 and t: D t = β (CH t ) ρ (C H t 1 ) ρ P t 1 P t (2.10) D t is defined analogously. S t is the nominal exchange rate at time t. The home government issues the local currency, has no expenditures, and runs a balanced budget every period. The nominal transfer is then given by: T t = M t M t 1 (2.11) The stochastic process that describes the evolution of the domestic money supply is: M s t = µ t M s t 1 (2.12) log(µ t ) = ε µ,t (2.13) where ε µ,t N(0, σ 2 ε µ ) is a normally distributed random variable. The stochastic process of money supply in the foreign country is entirely analogous. Also, the home monetary shock and the foreign monetary shock are assumed to be independently distributed, i.e., Cov(ε µ, ε µ) = Foreign Exchange Market Foreign Exchange Traders Following closely the work of De Long et al. (1990) and Jeanne and Rose (2002), the foreign exchange traders are modelled as overlapping generations of investors who decide how many one-period foreign nominal bonds to buy in the first period of their lives. Traders have the same taste, but differ in their abilities to trade in the foreign bond market. Some of them are able to form accurate expectations on risk and returns, while others have noisy expectation about future returns. The former are referred as the rational trader and the latter as the noise traders. Hereafter, the informed trader is denoted by a superscript I and the noise trader is denoted by a superscript N. Two specifications of the model are developed. In the first specification, the number of incumbent noise traders is exogenously determined. In the second one, the traders have to pay a fixed entry cost to trade on the foreign exchange market. The introduction of an entry cost 8

10 helps to endogenize the noise component of the market. This makes the policy analysis possible as policy makers can affect the composition of traders through the entry cost. In the foreign exchange market, at each period, a generation of foreign exchange traders is born. The continuum of the traders is indexed by i [0, 1]. Assuming that in each generation of traders, N I of them are rational traders, and 1 N I are noise traders. The timing of the model is illustrated in Figure 1. Figure 1: Timing of Model t t+1 Action 1 Action 2 Action 3 Action 1: Time t foreign exchange trader i is born; Time t shocks and nominal interest rates are revealed; The time t born trader i decides if he should enter the foreign bond market. Action 2: He decides the number of foreign currency bonds Bh,t+1 (i) to purchase based on his expectation about future exchange rate S t+1. To finance his purchase, he borrows B h,t+1 (i)s t from the home bond market. Action 3: Time t + 1 exchange rate S t+1 is revealed, so the return of his investment in terms of home currency is realized, which is equal to S t+1 Bh,t+1 (i)(1 + r t+1 ). He pays back the principle and interest of his borrowing(bh,t+1 (i)s t(1 + r t+1 )), gets the excess return, consumes, and dies. Let ϕ i t denote the dummy variable characterizing the market-entry condition of period t born foreign exchange trader i. If ϕ i t = 0, trader i will not enter the foreign bond market and if ϕ i t = 1, he will enter. At the beginning of period t, trader i will enter the market as long as the expected utility of entering the market is higher than that of not entering: E i t(u i t ϕ i t = 1) E i t(u i t ϕ i t = 0) (2.14) A foreign exchange trader who has entered the foreign bond market maximizes a mean- 9

11 variance utility function: 7 max Bh,t+1 (i) Ei t(ct+1(i)) T a 2 V ari t(ct+1(i)) T (2.15) Subject to C T t+1(i) = [B h,t+1(i)(1 + r t+1)s t+1 B h,t+1(i)s t (1 + r t+1 )] c i T T (i) (2.16) where Bh,t+1 (i) denotes the amount of one-period foreign currency bonds held by trader i from time t to time t + 1, a is the absolute risk aversion coefficient; c i reflects the costs associated with entering the foreign bond market for trader i. T T (i) is the trader i s transaction cost of trading foreign-currency bond. It is assumed that T T (i) = τ B h,t+1 (i)2 S t (2.17) 2 where τ > 0 is the rate of the transaction tax on foreign bond trading, or, in other words, the Tobin tax rate. Here, the transaction cost is modelled as a convex transaction cost because a linear cost would imply that trader i will gain when selling foreign bonds. 8 Note that the transaction cost is in foreign currency as trader i is trading foreign bonds. Therefore, in terms of domestic currency, it is τ B h,t+1 (i)2 2 S t. The entry costs may include information costs for investment in the foreign bond market and other costs when investing abroad. To formalize this heterogeneity, here we follow the specification used by Jeanne and Rose (2002). Rational traders are assumed to have a larger stock of knowledge about the economy and thus, do not need to invest in the acquisition of information. Their entry costs are therefore zero. For noise traders, they do not have a natural ability to acquire and process the information about the economy and therefore have to pay a positive entry cost. Although the preferences of all noise traders are the same, the noise traders are assumed to be distinguished from each other by their entry costs. Without loss of generality, the noise traders are indexed by increasing entry costs: { } i α c i = c 1 N I for i [0, 1 N I ] (2.18) 7 Ex post, the net return on the foreign exchange traders position could be negative. That is, the traders can have negative consumption. One way to fix this problem is to assume that each period traders enter the market with an endowment which is large enough to ensure their consumption can never be negative. So the foreign exchange traders will never default. Nevertheless, the existence of this endowment will not affect the optimal bond holding problem of the traders. Thus, we do not model it explicitly. 8 Also, in reality the transaction cost in foreign exchange market is usually convex in the amount of foreign currency traded. 10

12 where α > 0 is the curvature parameter and c is the parameter characterizing the scale or level of the entry cost of the noise traders. Thus, the noise trader at the left end of the continuum (i near 0) tends to have a lower entry cost and the noise trader towards the right end of the continuum (i near 1 N I ) has a higher entry cost Optimal demand for foreign bonds Once traders have decided to enter the market, the optimal demand for foreign bonds of each type of traders can be derived. Substituting Equations 2.16, 2.17 into Equation 2.15, gives: 9 { [ ] B max E i h,t+1 (i)s t (1 + r t+1 ) Bh,t+1 (i) t ρ t+1 c i τ B h,t+1 (i)2 S t a [ ]} B 2 2 V h,t+1 (i)s t (1 + r t+1 ) ari t ρ t+1 (2.19) where is the excess return. ρ t+1 = [ St+1 (1 + r t+1 ) S t (1 + r t+1 ) ] 1 (2.20) We now discuss the information structure of traders. Specifically, we make the following assumptions about the subjective distribution over ρ t+1. The rational traders can predict ρ t+1 correctly; while the noise traders cannot predict the future excess return correctly. That is, for informed traders: E I t [ρ t+1 ] = E t [ρ t+1 ] (2.21) V ar I t [ρ t+1 ] = V ar t [ρ t+1 ] (2.22) For noise traders, following the work of De Long et al. (1990), we assume: E N t [ρ t+1 ] = E t [ρ t+1 ] + v t (2.23) V ar N t [ρ t+1 ] = V ar t [ρ t+1 ] (2.24) V ar(v t ) = λv ar(s t ) where λ (0, + ) (2.25) where v t is assumed to be i.i.d and normally distributed with zero mean. λ can be considered as a parameter characterizing the relative magnitude of noise traders erroneous beliefs to exchange rate volatility. From Equations 2.23 and 2.21, it can be seen that, compared to the rational trader s expectation, the noise traders expectation of ρ t+1 based on time t information is biased from 9 Note that is known at t because we assume that price are set one period ahead. Meanwhile, B h,t+1, S t, and r t+1 are known at t by assumptions. 11

13 the true conditional expectation by a random error. Nevertheless, noise traders can correctly forecast the conditional variance of the exchange rate. From Equation 2.25, another assumption is made that the unconditional variance of v t is proportional to the unconditional variance of the exchange rate itself. This assumption helps to tie down the scale of the volatility of noise traders erroneous beliefs. 10 Solving Equation 2.19, the optimal bond holding of trader i is given by: B i, h,t+1 = Et[ρ i t+1 ] τ (1+r t+1 ) + a S t (1 + r t+1 )V ar t [ρ t+1 ] (2.26) Therefore, informed traders and noise traders differ in their optimal bond holding. Also, from Equation 2.26, the lower the expected excess return, the higher the risk (excess return volatility) and the risk coefficient, the less bond traders (both rational traders and noise traders) will hold. Thus, the traders account for risk when taking positions on assets. It can also be seen that the Tobin tax reduces the bond trading of both types of traders. This is quite intuitive, as foreign exchange traders will tend to trade less foreign currency bonds when there is a tax on transactions. At the margin, the return from enlarging one s position in an asset that is mispriced (the expected excess return) is offset by the additional price risk (the volatility of the excess return) and transaction cost (the Tobin tax) that must be borne Equilibrium condition of the foreign exchange market Analysis with no entry costs We first analyze a simple case where c = 0. As shown in Appendix A, if there is no entry cost, traders will always choose to enter the market. This is because the transaction cost is convex in the bonds traded, traders can always choose to hold a small amount of foreign bonds and get a positive expected utility, regardless of how large τ is. Thus, in this case all noise traders will enter the market and the noise component of the market is exogenously determined by the number of existing noise traders (1 N I ) on the market. So the aggregate demand for foreign bonds by foreign exchange traders of the home country can be denoted as: Bh,t+1 = N I B I, h,t+1 + (1 N I)B N, h,t+1 = E t [ρ t+1 ] + (1 N I )v t τ 1+r t+1 + a S t (1 + r t+1 )V ar t (ρ t+1 ) (2.27) 10 The logic behind this assumption is that the bias in noise traders expectation must be related to the volatility of the exchange rate itself, otherwise noise traders might expect the future exchange rate to be volatile even under a fixed exchange rate regime. 12

14 E t [ρ t+1 ] + (1 N I )v t a S t (1 + r t+1 )V ar t (ρ t+1 )B τ h,t+1 }{{} (1 + r t+1 ) B h,t+1 = 0 (2.28) F irst P art }{{}}{{} Second P art T hird P art Endogenous entry of noise traders We now endogenize the composition of traders who enter the market in each period by introducing positive entry costs for noise traders. The entry decision for informed traders is trivial. They bear no entry cost and always enter the foreign bonds market in equilibrium. A noise trader, however, enters if and only if Equation 2.14 is satisfied. As shown in Appendix A, for trader i, this condition takes the form: c i [Et N (ρ t+1 )] 2 P 2aV ar t (ρ t+1 ) + 2τ t+1 GBt N (2.29) S t(1+r t+1 ) 2 where GBt N is the gross benefit of entry for noise traders. It increases with the expected excess return and decreases with conditional time t + 1 exchange rate volatility and the Tobin tax. Let c t = GBt N be the cut-off value of entry cost. From Equation 2.29, for noise trader i, if c i c t, ϕ i t = 1; if c i > c t, ϕ i t = 0. The number of incumbent noise traders n t is then given by: [ (c ) 1 ] α n t = min t [E, 1 t N (ρ t+1 )] 2 (1 N I ) = min c P [2aV ar t (ρ t+1 ) + 2τ t+1 S t (1+r t+1 ] c ) 2 1 α, 1 (1 N I ) (2.30) Apparently, the number of active noise traders on the market increases with the square of the expected excess return and the number of existing noise traders, and decreases with the entry cost, the risk aversion coefficient a, the Tobin tax τ, and the excess return volatility. economic intuition behind Equation 2.30 is as follows. The presence of more active noise traders creates higher expected excess return and incentives for other noise traders to enter the market, however, the extra volatility brought about by their entry will reduce the gross benefit of entry for noise traders. In equilibrium, the two effects balance and no more noise traders will enter. Substituting Equation 2.30 into Bh,t+1 = N IB I, h,t+1 +n tb N, h,t+1, we can derive the equilibrium condition of the foreign bond market when the entry decision of traders is endogenized: E t (ρ t+1 ) + n tv t a S t(1 + r t+1 ) N I + n }{{ t P } t+1 (N I + n t ) V ar t(ρ t+1 )Bh,t+1 }{{} F irst P art Second P art where n t is given by Equation τ (1 + r t+1 )(N I + n t ) B h,t+1 }{{} T hird P art = 0 The (2.31) Equations 2.28 and 2.31 represent the interest parity conditions in this economy. Note that the uncovered interest parity does not hold in this model. The last three terms in Equations 2.28 and Equation 2.31 show the deviation from the uncovered interest parity when noise traders are present in the market. This deviation consists of three 13

15 parts. The first part is the expectation error of the noise traders, which increase when there are more incumbent noise traders. The second part is the risk premium term, since the foreign exchange traders are risk averse. Besides the expectation error term and the risk premium term, there is an extra term that comes from the transaction tax. Even in the absence of noise traders, the second and third term still exists Equilibrium Condition Equilibrium for this economy is a collection of 28 sequences (P t, P t, P h,t, P h,t, P f,t, P h,t, C t,c T t,c H t,c t, C h,t, C h,t, C f,t, C h,t, S t, r t, r t, D t, D t, W t, W t, B t, B t, B h,t, L t, L t, n t, ρ t+1 ) satisfying 28 equilibrium conditions. They include the six household optimality conditions (Equations 2.5, 2.6, 2.7 and their foreign counterparts), the definition of the price indexes (Equation 2.4 and its foreign analogy), the definition of the pricing kernel (Equation 2.10 and its foreign analogy), the interest parity conditions (Equation 2.28 or Equation 2.31), the definition of n t (Equation 2.30) for the endogenous entry case, the definition of excess return ρ t+1 (Equation 2.20), the four individual demand equations, the four pricing conditions, and the four market clearing conditions for the bonds and goods markets: Finally, the budget constraint of the foreign exchange traders. P t C T t = B h,t(1 + r t )S t B h,ts t 1 (1 + r t ) τ B h,t 2 B t+1 = S t B h,t+1 t (2.32) B t + B h,t = 0 (2.33) L t = C h,t + C h,t (2.34) L t = C f,t + C f,t (2.35) i=0 2 S t 1 Exogenous Entry (2.36) [ ] P t Ct T = Bh,t(1 + rt )S t Bh,tS n t t 1 (1 + r t ) P t c i τ B h,t 2 2 S t 1 Endogenous Entry (2.37) And the home country aggregate consumption equation: C t = Ct H + Ct T + τ B h,t 2 2 C t = Ct H + Ct T n t + c i + τ B h,t 2 2 i=0 S t 1 P t Exogenous Entry (2.38) S t 1 P t Endogenous Entry (2.39) 11 But as shown in the literature, these terms are not volatile enough to explain the exchange rate volatility. 14

16 Then the above two equations, the budget constraints of the home households 2.2, Equation 2.32 and its one-period lag can be combined to get the national budget constraint of the home country: where Π t = ω [ (P h,t W t ) P t C t = W t L t + Π t + S t B h,t(1 + r t ) S t B h,t+1 (2.40) ( Ph,t P t ) γ Ct + (P h,t S t W t ) ( P h,t P t ) γ C t ]. 3 Model Solution The detailed model solution, including the approximation of the system of equation and the derivation of solution, is given in the technical appendix. Here we will just outline the solution method. The equilibrium conditions of this economy can be divided into conditions with variance term V ar t (ρ t+1 ) (the interest parity conditions) and conditions without the variance term. Since the second-order terms are important for understanding dynamics of the economy, especially for the exchange rate, we will solve the model by the following steps. First, we will take a second-order approximation of the equilibrium conditions with variance terms: the interest parity conditions (Equation 2.28 and Equation 2.31). By second-order approximation, we can keep the variance term and expectation error terms. 12 Given the secondorder approximation of these equations, it can be shown that it is only necessary to solve a first-order approximation of the other equilibrium conditions of the model. 13 This is because the second-order approximations of Equations 2.28 and 2.31 only contain terms in the square and cross products of the variables of the model. Thus second order accurate solutions for these terms can be obtained from a first-order solution to the equilibrium conditions without variance terms. Therefore, the second step is to log-linearize other equilibrium conditions. The point of approximation is the non-stochastic, symmetric steady state (as described in Appendix B), where net foreign assets are zero, 14 all prices are equal, and the exchange rate is unity. Given 12 If we only use first-order approximation to approximate the interest parity conditions, then the expectation error terms and the variance terms will disappear. We could not analyze the endogenous entry case and the impact of Tobin tax. In other words, policy analysis is not possible. 13 This method is firstly used by Devereux and Sutherland (2006) to solve a model where there is portfolio holding. In our paper, the interest parity condition is analogous to their portfolio conditions. 14 The financial market are incomplete in our model since the home and foreign household only have access to non-state contingent domestic currency nominal bonds. If there are no foreign exchange traders, then there is a unit root in the net foreign assets in this kind of model. However, when the foreign exchange traders are present, the net foreign assets are zero at the steady state. Please see Appendix B for detail. 15

17 the approximated system of equilibrium conditions, the final step is to solve deviations of the exchange rate and other macroeconomic variables from their t 1 expectations in terms of exogenous money supply shocks and the expectation error shocks. exchange rate disconnect can be achieved in this model. And then we can see if Hereafter, ˆx t = log(x t ) log( X), dx t = X t X, where X is the non-stochastic steady state value of variable X t. And let from its date t 1 expectation. 15 x t+j = x t+j ˆ E t 1 ( x t+j ˆ ), j 0 denote the deviation of a variable 3.1 Model 1: Exogenous Entry Following the first two steps outlined above, we can get the approximated system of equations. Then, the log-linearized home household s budget constraint minus its t 1 expectation gives: 16 c H t c t + 2 P C db t+1 = s t (3.1) The right-hand side of Equation 3.1 represents the relative wealth effect of an unanticipated shock to the exchange rate through firms profits. This relative wealth increase will be spread between an increase in relative home consumption and net foreign assets accumulation. Using the log-linearized goods market clearing condition and Equation 3.1, we get: ( c H t c t ) + σ r E t( c H t+1 c t+1 ) = s t (3.2) where σ = 1 (1 γ)ρ 1+ψγ. This equation implies that the wealth effect of an increase in exchange rate will be spread between increase in current relative consumption and expected period t + 1 relative consumption. Then, using the log-linearized intertemporal optimality equations and the interest parity condition, we may obtain the consumption-based interest parity condition: E t ( c H t+1 c t+1 ) = ( c H t c t ) 1 ρ { s t (1 N I )v t + [ a (1 + r) S V ar t ( s t+1 ) + P τ 1 + r ] } dbh,t+1 This equation implies expected consumption growth in the home country decreases in response to an unanticipated exchange rate depreciation, since it generates an unanticipated real depreciation, and therefore reduces the home country s real interest rate. 15 Hereafter, the curvature parameter of entry costs α is set to be equal to 1. The model can be easily extended to the case where α > 1 or 0 < α < 1, and the main results will not change. Also, it is assumed that the elasticity of the money demand ɛ = 1, which is close to the estimate reported in Mankiw and Summers (1986). 16 See the technical appendix for details of derivations of equations. (3.3) 16

18 Finally, the relation between relative money supply and relative consumption can be derived from the money demand equations: 17 m t m t = ρ( c H t c t ) (3.4) Putting Equations 3.1, 3.2, 3.3 and 3.4 together, we can get a system of equilibrium conditions that characterizes { s t, c t c t, db h,t+1 }. monetary shocks. where This gives us the solution of s t in terms of v t and s t = ( m t m t )1 + σ r + φ ρ V ar t( s t+1 ) + ξ ρ τ σ ρ + σ r + φv ar t( s t+1 ) + ξτ + r ρ + σ r + φv ar t( s t+1 ) + ξτ (1 N I)v t (3.5) φ = a(1 + r) S C 2 σ r ξ = P C σ 2(1 + r) r (3.6) From Equation 3.5, the variance of the future exchange rate deviation, V ar t ( s t+1 ) can be solved. 18 Let V ar t ( s t+1 ) V s, V s is given by the following implicit function: V s = ( ) 1+ σ r + φ ρ V s+ ξ 2 ρ τ ρ+ σ r +φvs+ξτ ( σ ) 2 [V ar( m t ) + V ar( m t )] (3.7) r 1 ρ+ σ r +φv s+ξτ (1 NI ) 2 λ Note that the coefficient φ is associated with the risk-aversion of traders. The higher the risk aversion coefficient, the lower will be the exchange rate volatility. For both types of traders, their aversion to risk prevents exchange rate volatility from increasing too much. To see this, it can be ( ) 1+ σ r shown that as long as ρ > 1, holding other parameters constant, the numerator + φ ρ Vs+ ξ 2 ρ τ ρ+ σ r ( +φvs+ξτ σ ) 2 r is decreasing in V s and the denominator 1 ρ+ σ r +φv s+ξτ (1 NI ) 2 λ is increasing in V s. Similarly, it can easily be shown that given other parameters, the numerator decreases in τ, the rate of transaction tax, while the denominator increases in τ. So this implies that the Tobin tax will reduce exchange rate volatility. The exactly relationship between τ adn V s will be discussed in Section 4. Can the exchange rate display excess volatility in this model? When ρ = 1, the coefficient in front of ( m t m t ) is exactly 1 in Equation 3.5. Therefore, with no noise traders, the exchange rate volatility will be equal to that of the fundamentals. If noise traders are present on the 17 Notice that m t = ε µ,t, and m t = ε µ,t. We use m t and m t for notational convenience. 18 Since s t is linear in m t, m t and v t and the monetary shocks and expectation error shocks are normally distributed with zero mean and constant variance, V ar t ( s t+1 ) = V ar( s t+1 ) = constant V s. 17

19 market, the exchange rate volatility may be much higher than the fundamental volatility, even when ρ = 1. What are the responses of macroeconomic fundamentals such as consumption, labor and wage to the exogenous monetary shocks and expectation error shocks? From the log-linearized goods market clearing condition, labor supply condition and the money demand condition, w t = ψ 2ρ ( m t + m t ) + m t l t = 1 2ρ ( m t + m t ) (3.8) c t = 1 ρ m t c t = 1 ρ m t (3.9) Therefore, the volatilities of the macroeconomic fundamentals are only decided by the volatility of the relative monetary shock and consumer preferences, but not by the volatility of the expectation error and the number of incumbent noise traders in the market. Note that from Equations 3.1 and 3.4, the net foreign assets are given by: db t+1 = P C 2 [ s t 1 ρ ( m t m t )] (3.10) Thus, the volatility of the net foreign assets will be affected by the number of incumbent noise traders. But as shown in Section 4, their volatility is much smaller when compared to exchange rate volatilities. 3.2 Model 2: Endogenous Entry The endogenous entry case is similar to the exogenous entry case except for the interest parity equation. So following the same method, we can get the solution to the endogenous entry model, and the derivation is entirely analogous to Equation 3.5. The technical appendix gives details : φ s t = ( m t m + σ r + ρ t )1 V ar t( s t+1 ) + ξ ρ τ ρ + σ r + φ V ar t ( s t+1 ) + ξ τ + σ r ρ + σ r + φ V ar t ( s t+1 ) + ξ τ 1 N I n t v t (3.11) where n t min [E t ( s t+1 ) s t + v t ] 2 [2aV ar t ( s t+1 ) + 2τ P S(1+ r) 2 ] (1 N I ) c, 1 N I (3.12) φ = a(1 + r) S C 2N I σ r ξ = P C σ 2(1 + r)n I r (3.13) Analogous to the exogenous entry case, when ρ = 1 and no noise traders are present (n t = 0), s t = ( m t m t ), exchange rate volatility will be identical to that of the fundamental. When there are noise traders in the foreign exchange market, the exchange rate may diverge significantly 18

20 from the fundamental values. Note that now the Tobin tax not only affects the exchange rate volatility through the ξ τ term as in the exogenous entry case, but also affects the exchange rate volatility through n t, the number of incumbent noise traders. Therefore, the impact of the Tobin tax on exchange volatility depends on the market structure and the interaction of the Tobin tax with other trading cost on foreign exchange market. Section 4 will discuss the implication of the Tobin tax further. Finally, the expression for net foreign assets, consumption, labor, and wage are exactly the same as in the exogenous case. 4 Results Equations 3.5 and 3.11 are too complicated to be solved analytically, so the numerical undetermined coefficient method described in the technical appendix is used to solve for s t, V s and E s. Table 1 gives the parameter values that are used in the numerical simulation. We choose β = 0.94, which produces a steady state real interest rate of six percent, about the average longrun real return on stocks. The parameters η and ψ are set so that the elasticity of labor supply is 1 and the time devoted to work is one quarter of the total time in the steady state. The business cycle literature has a wide range of estimates for the curvature parameter ρ. Chari, Kehoe, and McGrattan (2002) set ρ = 5 to generate a high volatility of the real exchange rate. In our model, a high exchange rate volatility can be obtained without high risk aversion of households, so it is set to equal 2. For the final goods technology parameters, the elasticity of substitution between domestic produced goods θ is set to 11 following Betts and Devereux (2000). This gives a wage-price mark-up of about 1.1, which is consistent with the finding of Basu and Fernald (1994). The elasticity of substitution between home goods and foreign goods γ is set to be 1.5, following Chari, Kehoe, and McGrattan (2002) and Backus, Kehoe, and Kydland (1994). Note that, other parameters, such as the money supply process, number of informed traders on the market, and entry costs, are not fully calibrated as the purpose of this paper is just qualitative analysis. In the first subsection, we will analyze the implication of noise traders in explaining exchange rate disconnect. Then we will consider the endogenous entry of noise traders in the second subsection. We then explore the effect of the Tobin tax on exchange rate volatility in the last subsection. 19

21 TABLE 1 Parameter Values Exogenous Case Preferences β = 0.94, ρ = 2, ɛ = 1, η = 58.2, ψ = 1 Final goods technology θ = 11, γ = 1.5, ω = 0.5 Money Growth Process corr(ε µ, ε µ) = 0, σε 2 µ = σε 2 = 0.01 µ Foreign exchange traders c = 0, N I [0, 1], λ = 1.5 Steady State Values µ ss = µ ss = 1, M ss = Mss = 2 Endogenous Case a Foreign exchange traders c > 0, N I [0, 1], λ = 1.5 a Other parameters in the endogenous case are the same as in the exogenous case. 4.1 Exogenous Case We first solve for the exogenous entry case. Table 2 illustrates the results of simulations. The first ten rows show changes in volatilities of the exchange rate and net foreign assets when the number of noise traders increases from 0 to 1. The last three rows report volatilities of the macroeconomic fundamental variables, given the calibrated parameter values. From Table 2, three important findings are: First, the exchange rate volatility increases when the number of noise traders increases, while volatilities of macroeconomic fundamentals remain constant. Moreover, the exchange rate volatility is much higher than that of the macroeconomic fundamentals. Second, from the functional form of s t listed in Table 2, the impact of fundamental monetary shocks on the exchange rate (coefficient of m t or m t ) decreases in the number of noise traders. Meanwhile, the effect of expectation error on exchange rate (coefficient of v t ) increases when more noise traders are present on the market. Third, the exchange rate volatility is higher when the magnification coefficient λ increases. Therefore, the critical implication is that disconnection does exist between the exchange rate and the macroeconomic fundamentals in this model. The presence of noise traders in the foreign exchange market, combined with local currency pricing, generates a degree of exchange rate volatility that may be much higher than that of the underlying fundamental shocks. In other words, the exchange rate disconnect puzzle may be explained by the approach suggested in this paper. Why the disconnect puzzle can be explained in such a model? This is mainly because the presence of expectation errors and the risk premium term of noise traders drives wedges between 20